problem solving what is the number sentence

Number sentence – Definition, Application, FAQs, Examples

What is a number sentence, application of number sentences, solved examples on number sentence, practice problems on number sentence, frequently asked questions on number sentence.

A number sentence is a mathematical sentence made up of numbers and symbols, as shown below. 

The term “number sentence” is introduced at the elementary school level. However, the application of these sentences extends beyond elementary school because it includes equations and inequalities . These sentences can also be described as the language of mathematics. As shown below, a sentence combines two expressions with a relational symbol $(=, \gt, \lt, \text{etc.})$.

These sentences show the equality or inequality relations using different mathematical operations like addition , subtraction , multiplication , and division . 

The sign of equality and inequality is significant as the sentence is incomplete and makes no sense without them. 

$10 + 8 \gt 15$, is an example of a number sentence. However, if we write $10 + 8$  $15$, it does not make any sense.

A math sentence can be true or false depending on the information provided. 

A mathematical sentence that gives all the information and is known to be either true or false, as shown in the example below. 

Mathematical sentence problems can appear in the form of word problems, asking students how to write a number sentence.

For example: Mary has 10 strawberries. If Dan gives her 15 strawberries, how many strawberries does Mary have in total?

So, Mary has $10 + 15  = 25$ strawberries. 

Related Worksheets

Why Do Students Need to Be Fluent in Math Sentences?

• Mathematical sentences help students understand algebra. This involves weaving algebraic thinking into elementary and middle-school math.
• Math sentences provide flexibility to solve a problem as compared to basic algorithms. Using sentences, students can break the numbers out to see the value of each digit. They can compose and decompose numbers by place value or use other strategies, building their reasoning and mental math skills as shown in the example below.

Number sentences are simply the numerical expression of a word problem.

Example 1: Determine whether the following sentence is true or false.

$12 + 12 + 12 \lt 4 \times 12$

The expression on the right side of the inequality (less than) sign is $12 + 12 + 12$, which is equal to 36.

Solving expressions on the right side of the inequality (less than) sign, we get $4 \times 12$ or 48.

Since $36 \lt 48$, we can say the given sentence $12 + 12 + 12 \lt 4 \times 12$ is true.

Example 2: Complete the math sentence so that it is true.

$6 + 7 = 9$ $+$ $\underline{}$

$6 + 7 = 13$

So, to make the sentence true, $9$ $+$ $\underline{}$ must be equal to 13. Therefore, the missing number must be $13$ $–$ $9$ or 4.

Example 3: Substitute the value into the variable (x) and state whether the resulting sentence is true or false.

$12 –$ x $= 9$ , substitute 4 for x

If we substitute x as 4 in the given sentence, we $12$ $–$ $4 = 9$, which is false, as $12$ $–$ $4 = 8 ≠ 9$.

Example 4: Find the value of the x so that the following sentence is true.

$\text{x}$ $–$ $24 = 10$

Adding the same number to both sides of the equal sign will keep the sentence true.

To find the value of x, we can add 24 to both sides of the equal sign.

$\text{x}$ $–$ $24 + 24 = 10 + 24$

Therefore, $\text{x}$ $= 34$

Number sentence - Definition With Examples

Attend this quiz & Test your knowledge.

Which of the following is not a number sentence?

Select the correct statement for the sentences given below. $40 + 30 = 70$ $90 + 1000 = 1900$, identify the symbol that can fill the blank to make the sentence true 90 ◯ 20 = 70.

Is it important for a number sentence to be true?

A math sentence does not necessarily have to be true. However, every sentence gives us information, and based on the information provided, it is possible to change the statement from false to true.

What is the difference between equations and inequalities?

An equation is a mathematical sentence that shows the equal value of two expressions while an inequality is a sentence that shows an expression is lesser than or more than the other.

Can fractional numbers be written in the form of a number sentence?

Yes, fractional numbers can be written in the form of a sentence. For instance,

$\frac{3}{4}+\frac{5}{4} = \frac{8}{4}$

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Number sentences explained: Definitions and examples

What is a number sentence, what are the different types of number sentences, what is the difference between a true and false number sentence, how do you show that a number sign is not equal, what is a math sentence, are number sentences and ‘sums’ the same thing, why is understanding number sentences so important.

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Number sentences are one of the first things that primary school children are introduced to because it is one of the leading frameworks upon which most mathematics is taught.

Whether you’re solving simple addition problems or complex algebraic problems, you will utilise number sentences.

They help make sense of what calculations need to be done, and without them, you’ll just have numbers without a way to interpret them. But what exactly are number sentences?

That’s what we’ll explain in this article, along with the different types of number sentences, why understanding them is so important, and much more.

A number sentence can be defined as a mix of numbers and signs – also known as simple mathematical symbols – that presents a mathematical problem or equation which needs to be solved. As such, a number sentence is used synonymously with the phrases ‘math problem’ and ‘math equation’. Signs can be anything from operation signs such as multiplication (×) and division (÷) to an equal (=) or inequality (<>) sign.

Number sentences can include numbers and a mathematical operations sign on both sides and are often separated by an equals sign or inequality sign. 1 + 2 = 3 is considered a number sentence because it has all the necessary parts. It has numbers (1 and 2) and an addition sign (+), which is separated from another number (3) by an equal sign.

Due to the wide range of number and sign combinations to choose from, there are essentially an infinite amount of number sentences out there. However, they will typically fall into the following categories:

Addition number sentence

This is when the number sentence has an expression on one side, an equals sign, and then a number after it. To use the same example as above, 1 + 2 = 3 is an addition number sentence.

Subtraction number sentence

A subtraction number sentence usually follows the same format as an addition number sentence. An example of this would be 10 - 7 = 3.

Multiplication number sentence

Again, these follow the same format but with a multiplication sign instead. For instance, 4 × 4 = 16.

Division number sentence

The number sentence 6 ÷ 3 = 2 is a prime example of a division number sentence.

Less than number sentence

This is where the format changes. Instead of an equals sign, there will be a less than (<) sign. The less than sign shows that there is an imbalance in the number sentence, where the left expression is smaller than the right expression. An example of this would be 9 + 3 < 15

We know that 9 + 3 = 12. If we substitute 12 into the number sentence, we will get 12 < 15. This holds true since 12 is indeed less than 15.

Greater than number sentence

The same thing applies here, except we use a greater than sign. The greater than (>) sign shows that the left expression is bigger than the right expression. An example of this would be 20 + 3 > 21.

We know that 20 + 3 = 23. If we substitute 23 into the number sentence, we will get 23 > 21. This holds true since 23 is greater than 21.

Fraction number sentence

Fraction number sentences can include an equals sign or inequality sign but will have fractions instead of whole numbers. For instance, ⅕ + ⅗ = ⅘.

Algebraic number sentence

Lastly, we have algebraic number sentences. These substitute the whole numbers with letters such as a + b = c. As with fraction number sentences, these can vary, with some having equal signs and some having inequality signs.

A valid number sentence can be both true and false and is dependent on the expressions in the number sentence. Let’s explore what each one is and how they differ from each other.

True number sentence

A true number sentence is one where the written sentence is correct and is balanced on both sides. This is usually shown by using an equal sign to show that one side of the equation is equal to the other side of the equation. Examples of number sentences that are true include the following:

• 10 × 5 = 50
• 22 - 7 = 15

The above examples are relatively straightforward since they use the four main basic mathematical operators in an expression on one side, with the answer to the problem on the other. However, you can also have a true number sentence which consists of an expression on both sides. Calculating both expressions separately will result in equal values. For instance, suppose we have the number sentence 8 × 4 = 2 × 16.

If we solve this example problem, we will find that it is a true number sentence. 8 × 4 = 32 and 12 ×16 = 32, which we can write as 32 = 32. Since these are equal, it is a true number sentence. More examples of this are as follows:

• 9 - 8 = 1 ÷ 1
• 3 + 12 = 5 × 3
• 25 × 10 = 1,000 - 750

False number sentence

Whereas true number sentences are balanced on both sides of the equals sign, a false number sentence is one where it is unbalanced. This is often described as an ‘untrue’ problem. For instance, suppose we have the number sentence 5 × 4 = 15. If we calculate this problem, we will find that 5 × 4 is actually 20. 20 does not equal 15, and therefore this number sentence is defined as false.

False number sentences are typically used to test whether a student or person has a sound understanding of mathematical operations and expressions, as this deeper understanding will be required to distinguish between a false and true number sentence. More examples of false number sentences include the following:

• 25 × 9 = 750 ÷ 11

Sometimes, you will find that you don’t want to write a false number sentence, but you still need to show that the number sentence is not equal. This is where using an inequality sign makes sense.

There are four inequality signs to be aware of:

• Greater than (>) – The expression or values on the open side of the sign are said to be bigger than what’s on the closed side
• Less than (<) – The same holds true for this. The expression or values on the open side of the sign is said to be bigger than what’s on the closed side
• Greater than or equal to (≥) – The expression or value on the open side of the sign is said to be bigger or equal to what’s on the closed side
• Less than or equal to (≤) – The same holds true for this. The expression or value on the open side of the sign is said to be bigger or equal to what’s on the closed side

These signs provide much more flexibility since they show the relationship between both sides of an expression without having to create a false number sentence – particularly, the ‘... equal to’ signs as they can make all the difference in statistics and computer science problems where you can have unknown values. Here are a few examples of inequality signs being used in number sentences:

• 9 × 9 > 55
• 55 < 9 × 9
• 23 + 4 < 29
• n + 15 ≥ 20: If you calculate this problem, it means that n must be equal to or greater than the value 5.

So far, the examples we’ve seen of number sentences have been written as numbers and symbols. But, you can also describe number sentences as math sentences, also known as written word problems.

This is where the problem is described using words instead of explicit operational signs. Many schools use math sentences to train their students on how to use their understanding of language to create a number sentence which they can then solve. An example of this would be the following sentence:

Jack has 4 apples, and Liz has 10. If Liz gives 3 apples to Jack, how many apples does she have left?

To calculate this, students would have to connect the dots and translate the words into a number sentence with numbers and operational signs. This particular problem can be solved by doing 10 - 3 = 7. The answer to this problem would be 7 apples.

In the past, teachers and educators would use the word ‘sum(s)’ when referring to number sentences. The problem with this was that sum has different meanings depending on its context.

For instance, the sum is used synonymously with the word ‘total’; you could say ‘the sum of 1 + 1 = 2.' Therefore, referring to number sentences as a sum can confuse those who are new to learning mathematics.

To correct this, in most English-speaking countries such as the USA, UK, Canada, Australia, and New Zealand, math problems are now described as number sentences – or alternatively, problems and equations. This means that when number sentences are mentioned, students immediately know what is being referenced.

Understanding how to write and use number sentences is crucial because it displays fundamental mathematical structures and forms the basis of algebraic equations, which are a key part of any curriculum.

Number sentences also allow students to learn the correct syntax, which is akin to learning the proper punctuation and grammar when writing in English. Without knowing how to write and interpret number sentences, the why behind the calculations remains unclear and can significantly hinder their math progress. This ability to record and analyze connections between numbers is an invaluable method to further your mathematics knowledge.

The benefits of learning how to use number sentences aren’t limited to the classroom. They also have real-life applications. Whether you’re in a bank looking to withdraw money or simply calculating how many ingredients you need to buy for a two-person meal, number sentences will be the framework that you use to solve the problem.

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What Is A Number Sentence: Explained For Primary Parents And Kids!

Sophie Bartlett

In this post, we will be answering the question “ what is a number sentence?” and running through everything you need to know about this particular part of primary maths. We’ve also got number sentence questions and worksheets that you can use to test out your child’s skills.

What is a number sentence?

A number sentence is a combination of numbers and mathematical operations that children are often required to solve. 

Example of a number sentences include:

32 + 57 = ?  

5 x 6 = 10 x ?

103 + ? = 350

They will usually comprise of addition, subtraction, multiplication or division – or a combination of all four!

Remember – You may consider the above simply as “sums”, but referring to them as this can be confusing for children because the word “sum” should only be used when discussing addition. 

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When will my child learn about number sentences?

In the English National Curriculum, number sentences are referred to as ‘mathematical statements’.

These math sentences or statements are introduced as a maths skill in Year 1 , where pupils read, write and interpret mathematical statements involving mathematical symbols including addition (+), subtraction (–) and equals (=) signs.

Number sentences build on what children will have already learnt about number bonds .

Then, children will expand on this:

• Year 2 pupils calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals sign (=).
• Year 3 pupils write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods. These pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency.
• Year 4 pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4).
• Year 5 pupils are expected to understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 81 x 10). 
• They should also recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number [for example, 2/5 + 4/5 = 6/5 = 1 and 1/5].
• Year 6 pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.

So, for example, in Year 1 your child will begin with addition sentences and subtraction number sentences. A number line may be helpful at this stage. In Year 2 they will use division sentences using whole numbers. By Year 4, they will use decimals in their number sentences.

Wondering about how to explain other key maths vocabulary to your children? Check out our Primary Maths Dictionary , or try these other maths terms:

• What Is The Perimeter?
• What Is BODMAS (and BIDMAS)?
• Properties of shapes
• What are 2D shapes ?
• What are 3D shapes ?

Number sentence practice questions

1) Complete the number sentences.

340 ÷ 7 = ____  remainder ____                          

____÷ 3 = 295 remainder 2

2) Here is a number sentence.

____ + 27 > 85

Circle all  the numbers below that make the number sentence correct.

30           40           50           60           70

3) Write in the missing number.

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Number sentence

A number sentence is a "mathematical sentence" used to express various mathematical relationships, namely equality and inequality. Number sentences are made up of:

• Operations (addition, subtraction, multiplication, division, etc.)
• Equality / inequality symbols

Below are some examples of number sentences.

Number sentences can also be written with fractions, decimals, negative numbers, with powers, and more. We can identify all the examples above as equations based on the use of the "=" sign. It is worth nothing that number sentences do not necessarily have to be true. For example, 2 - 3 = 5 is still a number sentence, albeit a false one. False number sentences can be used to test our understanding of basic arithmetic and all the symbols involved. For example, one thing we could change about the false number sentence is the minus sign. If we changed the minus sign to a plus sign, the number sentence would be true:

Number sentences can also take the form of inequalities . The key inequality symbols that we should recognize are:

• less than: <
• greater than: >
• less than or equal to: ≤
• greater than or equal to: ≥
• is not equal to: ≠

In the false number sentence above, 2 - 3 = 5, instead of changing the minus sign, we could also instead use various inequality signs. 2 - 3 = -1, so we could also write 2 - 3 < 5, and the number sentence would be true. Alternatively, we could write 2 - 3 ≠ 5, and this would also be true.

Change the following false number sentences such that they become true.

1 . 2 + 8 < 10:

2 + 8 ≤ 10

2 + 8 ≥ 10

2 . 5 ≥ 12:

2.2.2 Number Sentences

Standard 2.2.2.

Understand how to interpret number sentences involving addition, subtraction and unknowns represented by letters. Use objects and number lines and create real-world situations to represent number sentences.

For example : One way to represent n + 16 = 19 is by comparing a stack of 16 connecting cubes to a stack of 19 connecting cubes; 24 = a + b can be represented by a situation involving a birthday party attended by a total of 24 boys and girls.

Use number sentences involving addition, subtraction, and unknowns to represent given problem situations. Use number sense and properties of addition and subtraction to find values for the unknowns that make the number sentences true.

For example : How many more players are needed if a soccer team requires 11 players and so far only 6 players have arrived? This situation can be represented by the number sentence 11 -  6 = p or by the number sentence 6 +   p = 11.

Standard 2.2.2 Essential Understandings

Second graders build on their previous work with operations by writing number sentences to represent a real world or mathematical situation involving addition and subtraction. In addition, they write a real world problem to represent a given number sentence.

Work with number sentences continues as second graders determine if a number sentence is true or false. For example, is 9 + 7 = 14 true or false? They are also able to write their own true number sentences and false number sentences.

Second graders build on their understanding of the equal sign as they continue work with variables when finding unknowns in number sentences. The unknowns in given number sentences are found in varying positions. For example, 5 + k = 14, 5 = m - 9, 14 - r = 5.  Second graders understand that a variable used more than once in a given number sentence has the same value and different variables in a given number sentence may or may not have the same value. For example, in the equation r + r = 18, "r" equals 9.  In the equation a + b = 18, "a" and "b" could have the same value (9) or different values (10 and 8 or 12 and 6, etc.)

All Standard Benchmarks

2.2.2.1 Understand how to interpret number sentences involving addition, subtraction and unknowns represented by letters.  Use objects and number lines and create real-world situations to represent number sentences. 2.2.2.2 Use number sentences involving addition, subtraction, and unknowns to represent given problem situations.  Use number sense and properties of addition and subtraction to find values for the unknowns that make the number sentences true.

What students should know and be able to do [at a mastery level] related to these benchmarks:

• Know the equal sign means "the same as"
• Solve equations with the unknown in any position:

24 + 79 = n; n = 24 + 79 24 + n = 103; 103 = 24 + n n + 79 = 103: 103 = n + 79 103 - 79 = n; n = 103 - 79 103 - n = 24; 24 = 103 - n n - 79 = 24; 24 = n - 79 20 + 52 = n + 23

• Model the above types of equations using representations including manipulatives and number lines
• Create number sentences to match given story problems
• Create story problems to match given number sentences
• Determine the truth value of number sentences
• Demonstrate an understanding of the relationship between addition and subtraction

Work from previous grades that supports this new learning includes:

• Represent real-world situations involving addition/subtraction basic facts using objects and number sentences.
• Create real-world situations corresponding to number sentences.
• Determine if number sentences involving addition/subtraction are true.
• Use number sense and models to identify the missing number in an number sentence,
• Understand basic fact families (current grade level).

NCTM Standards

Represent and analyze mathematical situations and structures using algebraic symbols.

Pre-K - 2 Expectations

• Illustrate general principles and properties of operations, such as commutativity, using specific numbers;
• Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

Use mathematical models to represent and understand quantitative relationships.

• Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols. (NCTM, 2000)

Common Core State Standards

Represent and solve problems involving addition and subtraction.

• 2.OA.1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Misconceptions

Student misconceptions and common errors.

 Students may think...  

• the only format for a problem is a + b = c or a - b = c, not recognizing that it can also be c = a + b or c = a - b.
• n = 90 - 13 is read: 13 minus 90 equals n. Students need constant reminders to read left to right on either side of the equal sign.
• they can ignore the presence of letters or unknowns in an equation.
• there are rules that determine which number a letter stands for. For example, e = 5 because e is the fifth letter of the alphabet or y = 4 because y was 4 in the last number sentence.
• that a letter always has one specific value.
• different letters always represent different numbers.
• an equal sign means "and the answer is." In this way, when they see an equal sign, they want to carry out the operation preceding it. They need to think of the equal sign as meaning "is the same as."

In the Classroom  

Throughout the school year, the second graders in Mr. O's class have been working on solving story problems with a variety of contexts, using bar models to represent problems, solving problems with the unknown in various positions and writing equations to match story problems.

Mr. O. knows that different structures in problems can provide differentiation for his students. Problems where the result of the operation is unknown, is the easiest type of problem to solve - and in this case - write, because children have a great deal of experience with this type of problem. The change unknown problems are harder for children to solve. Problems that have the beginning quantity unknown are the most difficult for children to solve because it's challenging to figure out where to begin.

Mr. O.: Who's ready to write some story problems? Today we're going to build on activities we've been doing all year long. I'm going to give you some equations and you're going to write story problems to match them. What is an "equation?"

Marion:It's a number sentence with an equal sign in it.

  Mr. O.:Thanks, Marion. That's a good definition. Here are three equations. What does "n" represent in these number sentences?

57 -  29  =   n

57 -   n  =  28

 n -  29  = 28

Tyler: It stands for some number. We don't know what it is - that's what we have to figure out.

Mr. O.: Nice, Tyler. What do you notice about the equations on the board?

Paul: Well, they're like a fact family.

Mr. O.: Tell me more.

Paul: I already know what the "n" stands for because the answer is right there in the three number models. See, the first two problems have the total number. Then the next problem has the other two numbers, 27 and 29. So, I know all the numbers I need - 56, 27, and 29.

Mr. O: Nice connection, Paul. The point of this lesson is to write story problems that match the equations - now you have hints for solving your problems after you write them! You're still going to have to show your work and how you came up with the equations to match your story problems.

You need to choose the unit for your problem and you can choose the equation you want to work with.

Jose: Can we do all three?

Mr. O.: Sure, but I want you to work on one equation at a time. When you've finished writing the story problem, and solving it, you may work on another equation. If you're going to work on another equation, I would like you to use the same context and just change the location of the unknown.

Alaina: Can I work with LaMya?

Mr. O.:That's fine. I want you each to write different story problems. You can even choose different equations. LaMya will help you figure out how to write yours, if you have any questions.

The class gets to work. The children are used to working collaboratively and talking while they work. They know how to help each other by asking questions and not telling each other what to do. The children have solved many story problems during the year and they have experience choosing the equations that challenge them but still have an entry point so they can be successful. Mr. O. interacts with students as they work.

Mr. O.:Ahlam, what unit would you like to use in your story problem?

Ahlam: Puppies.

Mr. O.: Okay, look at the number model you chose. Read it to me.

Ahlam: 57 minus 29 equals n.

Mr. O.: Read it another way.

Ahlam: 57 take away 29 equals n.

Mr. O: Is there another way to read it?

Ahlam: 57 subtract 29 equals n.

Mr. O.: What else can we say for this symbol (points to equal sign)?

Ahlam: Same as. It means it is the same as. 57 minus 29 is the same as n.

Mr. O.: Nice. So, how is your story problem going to start?

Ahlam:There were 57 puppies.

Mr. O.: That's a good start. Since this is a subtraction problem, what kind of action are you going to use in your story?

Ahlam: Well, they could get adopted.

Mr. O.:Okay, write that down.

Ahlam wrote,"There were 57 puppies 29 found new homes."

Mr. O.:Remember your story needs a question. What question are you going to ask?

Ahlam:How many is left?

Mr. O: Will there be more than 57 puppies or fewer than 57 puppies left?

Ahlam: Less because it's a take away problem.

Mr. O.: Will there be more than 10 puppies or fewer than 10 puppies left?

Ahlam shrugs her shoulders.

Mr. O.: Make a ball park estimate.

Ahlam: Well, 57 is between 50 and 60, but closer to 60, and 29 is almost 30. So, it would be 60 take away 30.

Mr. O.: How much would that be?

Ahlam: It's like 6 take away 3 only it's about tens, so 6 tens take away 3 tens is 3 tens which is 30.

Mr. O.: So, your answer will be about 30. Is that more than 10 or less than 10?

Ahlam: It's more than 10.

Mr. O.: Right, so please write How many puppies 'are' left, because it needs to be a plural verb. Then draw a model to show how to set your solution up.

Mr. O.moves to another table.

Lisa: Mr. O., how do you spell "dolphins?

Mr. O.: Write down the sounds that you hear, and for the "f" sound use "ph."

Lisa: Okay.

Mr. O.: Which equation are you using?

Lisa: n minus 29 is 28.

Mr. O.: Good choice. What does the "n" stand for?

Lisa:That's how many dolphins there were in the beginning.

Mr. O.:So, how are you going to start your story problem?

Lisa:There were "some" dolphins by the boat.

Mr. O.: Good start. Keep working .

Miko: I'm going to write about limos.

Mr. O.: Fun - Which equation are you going to use?

Miko: 57 minus n equals 28.

Mr. O.: What does the "n" stand for?

Miko: It's some number that is being taken away.

Mr. O.:What is going to happen with your limos?

Miko: They're going to break down.

Mr. O.: That makes sense. Are there going to be more or fewer than 57 limos that break?

Miko:Less because 28 of them will still work.

Mr. O.: Will there be more than 20 or fewer than 20 that break?

Miko: I think it will be close to 30.

Mr. O.: Why?

Miko: Because I'm starting with about 60 and there are about 30 left, so that means about 30 are broken.

Mr. O.:Sounds good, keep working.

Mr. O. went back to Ahlam to check on her progress. 

Mr. O.: How did you figure out your answer?

Ahlam: I used an empty number line.

Mr. O.: How did you set it up?

Ahlam: Well, I put 29 here (on the left) and 57 here (on the right).  Then I started at 29 and counted up to 57. Then I added all my jumps. The answer is 28. So, there is 28 puppies left. 

Mr. O.: There are 28 puppies left. Your work is pretty clear and I can follow what you did.  Good job writing your answer in a sentence!

Lisa: Can I solve another problem now?

Mr. O.: Show me what you've done so far.

Mr. O.: Interesting. It seems like this is a subtraction problem. Why did you use addition to solve it?

Lisa: Because it's like a fact family and I know the biggest number is missing, so I have to plus the two small numbers. Adding is the opposite of minusing, and I'm missing part of my take away problem so I used adding to find it.

Mr. O.: Good thinking, Lisa. Go ahead and choose another equation to work with. Remember to still write about dolphins.

Miko: I'm finished with my story problem.

Mr. O.: It's nicely laid out, Miko. If your original equation was 57 minus n is 28, why did you write 57 minus 28?

Miko: Because I know I started with 57 limos. Some broke down, so I had 28 left. But to find out how many broke down, I had to take all the limos and take off the ones that are still working - then I can find out how many were broken. It shows it on the bar model - I'm missing a part so I can subtract the part I know.

Mr. O.: How did you figure out the answer?

Miko: I used an empty number line to add up.

Mr. O.: That's a clear explanation. Thanks, Miko.

Lisa, how is your second problem coming?

Lisa: I'm done.

Lisa:I used dolphins again.

Mr. O.: I notice that your number model matches the structure of your story problem. Then you solved it in a way that follows your number model. Nice work, Lisa!

Class, let's come back together and share some of our story problems before our time is up.

Mr. O. had several students share their story problems, with other students coming up to the board to write the matching number models, to make sure the children can go both ways - writing equations to match situations and devising situations to match equations.

Teacher Notes

• Students may need support in further development of previously studied concepts and skills .
• Research has shown that young children can use letters to represent unknowns. The number sentence 10 - s = 6 can be read 10 minus a number equals 6.
• Students need the opportunity to solve equations with the unknown in various locations. For example: 48 - m = 39, r + 15 = 38, 58 + 39 = k
• Students need to see the equal sign as a symbol of a relationship. They need to develop a broader algebraic understanding in which the expression on the left-hand side of the equals sign represents the same quantity as the expression on the right-hand side of the equals sign. They need to think of the equals sign as meaning "is the same as" instead of "the answer comes next."
• It is not necessary for teachers to teach commutativity or associativity as vocabulary words, but to on build students' understanding of the properties in observing mathematical situations and recording thinking. Students should be encouraged to investigate whether specific observations and hypotheses hold true for all cases (NCTM, 2000).
• Second graders use their number sense and knowledge of the properties of addition and subtraction in solving for an unknown. In solving 7 - m = 2, ask students what subtraction fact is related to this subtraction fact. 7 - 2 = 5. Ask, "Knowing that fact, what number is the unknown in 7 - r = 2? (5) Students may use the strategy of "counting on" to solve for the unknown. In thinking about 5 + k = 8 or 8 - 5 = k, children "count on" to find how many more it takes to get to 8.
• Modeling word problems is critical as students develop an understanding of operations and the relationships that exist between and among those operations. The bar model is an effective tool that can be used to represent the relationships in a variety of problem solving situations.  
• Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started What do you need to find out? What do you know now? How can you get the information? Where can you begin? What terms do you understand/not understand? What similar problems have you solved that would help? While Working How can you organize the information? Can you make a drawing (model) to explain your thinking? What are other possibilities? What would happen if...? Can you describe an approach (strategy) you can use to solve this? What do you need to do next? Do you see any patterns or relationships that will help you solve this? How does this relate to...? Why did you...? What assumptions are you making? Reflecting about the Solution How do you know your solution (conclusion) is reasonable? How did you arrive at your answer? How can you convince me your answer makes sense? What did you try that did not work? Has the question been answered? Can the explanation be made clearer? Responding (helps clarify and extend their thinking) Tell me more. Can you explain it in a different way? Is there another possibility or strategy that would work? Is there a more efficient strategy? Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

NCTM Illuminations

• Balancing Act 

Problems such as those in this activity help develop what students already know in preparation for writing equations and learning ways to solve for variables. Students use mathematical models to explore quantitative relationships. When presented with pictures of pan balances with one or more objects in each pan, they communicate relationships between the weights of the objects by comparing the balanced and unbalanced pans.

• In Balance  

In this lesson, students have an opportunity to explore the foundations for equivalence, an important step in their development of algebraic thinking as they see how quantities relate.  Students explore equivalence by comparing weights of different collections of objects.

• Block Pounds  

Students explore the use of variables as they solve for the weights of objects using information presented in pictures. They model situations that involve adding and subtracting whole numbers, using objects, pictures, and symbols.

• Finding the Balance

This lesson encourages students to explore another model of subtraction, the balance. Students will use real and virtual balances. Students also explore recording the modeled subtraction facts in equation form.

Instructional Activities

• True or False

Asking second graders to determine if a number sentence is true or false is important as second graders develop a sense of equality. It is important to use number sentences involving addition/subtraction of familiar numbers. Number sentences such as 14 = 14 and number sentences which are false (9  + 8 = 19), challenge student thinking. It is also important for second graders to consider the following types of number sentences:            

6 + 7 = 6 - 7, 9 + 7 = 7 + 4, 5 + 8 = 5 + 8,  16 = 9 + 7,  7 = 15 - 8.

The following group of number sentences might be used as second graders are asked to determine if a number sentence is true or false.

True or False?

6 + 7 = 13 7 + 6 = 13 13 = 6 + 7 6 + 7 = 14 13 = 13 13 = 7 + 7 6 + 7 = 6 + 7 6 + 7 = 7 - 6 6 + 7 = 6 + 6 + 1

Additional Instructional Resources    

Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Lubinski, C. A., and Otto, A.D. (2009). Literature and algebraic reasoning.   In Algebraic Thinking, Grades K - 12: Readings from NCTM's School-Based Journals and Other Publications, 99 - 105. Reston, VA: National Council of Teachers of Mathematics.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.   New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.)  Boston, MA:  Allyn & Bacon.

Van de Walle, J. & Lovin, L.  (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Von Rotz, L., & Burns, M. (2002). Grades k-2 lessons for algebraic thinkng. Sausalito, CA: Math Solutions Publications.

Willoughby, S. S. (1997, February). Functions from Kindergarten through Sixth Grade. Teaching Children Mathematics, 3, 314 - 18.

Unknown :  The number that is not known.

"Vocabulary literally is the

 key tool for thinking."

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions.  Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

Integration : Connecting new vocabulary to prior knowledge and previously learned vocabulary.  The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

Repetition :    Using the word or concept many times during the learning process and connecting the word or concept with its meaning.  The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

Meaningful Use:   Multiple and varied opportunities to use the words in context.  These opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems.  Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.

Strategies for vocabulary development

Students do not learn vocabulary words by memorizing and practicing definitions . The following strategies keep vocabulary visible and accessible during instruction.

Mathematics Word Bank:  Each unit of study should have word banks visible during instruction.  Words and corresponding definitions are added to the word bank as the need arises.  Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.

Labeled pictures and charts:  Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.

Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.

Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words.  Teachers should use these during the instructional process to engage student in thinking about the meaning of words.

Vocabulary Strips:  Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.

Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.

Additional Resources for Vocabulary Development

Murray, M. (2004). Teaching mathematics vocabulary in context . Portsmouth, NH: Heinemann.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Reflection - Critical Questions regarding the teaching and learning of these benchmarks

What are the key ideas related to an understanding of equality at the second grade level?  How do student misconceptions interfere with mastery of these ideas?

What kind of number sentences should second graders see related to equality in an instructional setting?

Write a set of number sentences you could use with second graders in exploring their understanding of equality. Which are the most challenging for second graders?

When checking for student understanding of equality, what should teachers

• listen for in student conversations?
• look for in student work?
• ask during classroom discussions?

Examine student work related to a task involving equality. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

What are the key ideas related to identifying an unknown in an equation at the second grade level? How do student misconceptions interfere with mastery of these ideas?

What kind of equations should second graders experience when identifying an unknown in an equation?

Write a set of equations you could use with second graders in exploring their understanding of identifying unknowns in equations. Which are the most challenging for second graders?

When checking for student understanding of solving for an unknown, what should teachers

Examine student work related to a task involving identifying unknowns in equations. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

How can teachers assess student learning related to these benchmarks?

How are these benchmarks related to other benchmarks at the second grade level?

Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K.. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Blanton, M. (2008). Algebra and the elementary classroom, transforming thinking, transforming practice.   Portsmouth, NH: Heinemann.

Carpenter, T., Franke, M., & Levi, L. ( 2003). Thinking mathematically integrating arithmetic & algebra in elementary school , Portsmouth, NH: Heinemann.

Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach, grades k-8 , 2nd edition . Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6) . Sausalito, CA: Math Solutions.

Fosnot, C. T. & Jacob, B. (2010). Young mathematicians at work:  Constructing algebra. Portsmouth, NH: Heinemann and Reston, VA: NCTM.

Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding.  Portsmouth, NH: Heinemann.

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Burns, Marilyn. (2007). About teaching mathematics:  A k-8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.

Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Caldera, C. (2005). Houghton Mifflin math and English language learners .  Boston, MA:  Houghton Mifflin Company.

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.     

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, Grades K-8 . (2nd ed.). Sausalito, CA: Math Solutions Press.

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2.   Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2 . Reston, VA: National Council of Teachers of Mathematics.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course . Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

Hyde, Arthur. (2006) Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders . Portsmouth, NH:  Heinemann.

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Murray, M., & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann .

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions .

Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.

Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades  K-3. Boston, MA: Pearson Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann .

4 + m = 16 Which number is equal to m? A. 10 B. 12 C. 11 D. 6   Solution:           B. 12 Benchmark:      2.2.2.1

• Crystal and Dominic picked 73 strawberries altogether. Crystal picked 28 strawberries. Which number sentence matches the story?

A. 73 + 28 =  s B. 28 + s = 73 C. 73 - 45 =  s D. 28 -  s =  73 Solution:           B. 28 + s = 73    Benchmark:      2.2.2.2

Riannon had 15 beads this morning.  Her mom gave her 24 more. How many beads does she have in all?  Write a number sentence to represent this story problem and answer the question, "How many beads does she have in all?" Solution:           15 + 24 = m,   39 beads Benchmark:      2.2.2.2

• There are twenty-five students in Mrs. Eischen's class. Twelve of the students went to library. How many students did not go to the library?  Write a number sentence that can represent this situation and solve it.

Solution:          25 - 12 = n,  n = 13 Benchmark:      2.2.2.1

• A soccer team requires 11 players and so far only 6 players have arrived. 

How many more players are needed?  Write a number sentence with a variable for the unknown. Answer the question, "How many more players are needed?" Solution:   6 + n = 11,  5 more players are needed (n = 5) Benchmark: 2.2.2.2

• Write a story to go with this number sentence:  42 - n = 28

Solve the number sentence for n. Solution:          Story corresponds to the given equation, n = 14. Benchmark:     2.2.2.2

Differentiation

More practice in looking for the related addition or subtraction fact may be necessary. Write a number sentence such as 2 +  n  = 8.  Have students show the whole (8) with colored counters, then separate (do not remove) 2 counters to make the known part. Ask, "How many counters are left to make the missing part?" (6). If the student is ready, use the markers to illustrate the relationship between 2, 6 and 8 (Fact Family). Once students understand the relationship between addition and subtraction they can work with larger numbers.

Concrete - Representational - Abstract Instructional Approach

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.

The CRA approach is based on three stages during the learning process:

              Concrete     -     Representational     -     Abstract

The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts.  At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level.  Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task.   Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.

The Representational Stage is the drawing stage.  Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems.  They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking.  Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.

The Abstract Stage is the symbolic stage.  Teachers model mathematical concepts using numbers and mathematical symbols.  Operation symbols are used to represent addition, subtraction, multiplication and division.  Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding.  Moving to the abstract level too quickly causes many student errors.   Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations. 

Additional Resources

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann .

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

English language learners find word problems much more challenging than symbolic math problems. The English language is very complex, with numerous nuances that must be learned. Students need to be able to translate common words to math symbols, natural language to algebraic expressions, and algebraic expressions to natural language.  Provide extra support for ELLs as they work through the complexity of language.

Always go back to the unknown and ask, "What does the unknown represent?  Have them explain what the "m" or "t" mean in a problem. A context increases the level of student understanding.

• Word banks need to be part of the student learning environment in every mathematics unit of study.
• Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.

• Sentence Frames

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions.  Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks:

• When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding. 

Additional ELL Resources:

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications .

Students expand their understanding of equality by writing their own true or false number sentences on slips of paper and placing them in a container. After drawing a slip of paper from the container, students determine if the number sentence is true or false.

Students can write open number sentences on slips of paper and place them in a container. After drawing a slip of paper from the container, students tell or write a number story to match the open number sentence and then find the unknown.

Students can determine unknown numbers on magic squares . 

Bender, W. (2009). Differentiating math instruction-strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom-a guide for teachers k-8. Portsmouth, NH: Heinemann .

Small, M. (2009). Good questions: great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

Parents/Admin

Administrative/peer classroom observation.

What should I look for in the mathematics classroom? ( Adapted from SciMathMN,1997)

What are students doing?

• Working in groups to make conjectures and solve problems.
• Solving real-world problems, not just practicing a collection of isolated skills.
• Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
• Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
• Recognizing and connecting mathematical ideas.
• Justifying their thinking and explaining different ways to solve a problem.

What are teachers doing?

• Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
• Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
• Connecting new mathematical concepts to previously learned ideas.
• Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
• Selecting appropriate activities and materials to support the learning of every student.
• Working with other teachers to make connections between disciplines to show how math is related to other subjects.
• Using assessments to uncover student thinking in order to guide instruction and assess understanding.

For Mathematics Coaches

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8 . (2nd ed.).  Sausalito, CA: Math Solutions.

Sammons, L., (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

For Administrators

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC : National Academies Press.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Parent Resources

Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc

Helping your child learn mathematics Provides activities for children in preschool through grade 5.

What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN.

Help Your Children Make Sense of Math

Ask the right questions

In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do. Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started What do you need to find out? What do you know now? How can you get the information? Where can you begin? What terms do you understand/not understand? What similar problems have you solved that would help?

While Working How can you organize the information? Can you make a drawing (model) to explain your thinking? What are other possibilities? What would happen if . . . ? Can you describe an approach (strategy) you can use to solve this? What do you need to do next? Do you see any patterns or relationships that will help you solve this? How does this relate to ...? Can you make a prediction? Why did you...? What assumptions are you making? Reflecting about the Solution How do you know your solution (conclusion) is reasonable? How did you arrive at your answer? How can you convince me your answer makes sense? What did you try that did not work? Has the question been answered? Can the explanation be made clearer? Responding (helps clarify and extend their thinking) Tell me more. Can you explain it in a different way? Is there another possibility or strategy that would work? Is there a more efficient strategy? Help me understand this part...

Adapted from They're counting on us, California Mathematics Council, 1995.

Algebra is a natural extension once kids have an understanding of counting, adding, and subtracting. At this level, your child is simply solving a number sentence with an unknown number. You can present your child with real-world situations and encourage them to use objects and/or write a number sentence to solve them.  These are some examples:

"If I have $14 in my pocket and pay Tom and I am left with $9, how much did I pay him?"  (The number sentence would be $14 - d = $9.  Any letter or symbol can be used for the unknown number.  Your child can use their knowledge of the related subtraction fact 14 - 9 = 5 to help them find the unknown number.  Your child might show 14 dollar bills to start and remove 9 to find 5 are left. So, d=5 and I paid Tom $5).

"Summer has three pieces of gum in one pocket and five in the other. How many altogether? (3 + 5 = g, 3 + 5 = 8)  If she has three in one pocket but eleven total how many in this pocket?" (11 - 3 = s or 3 + s = 11)

Related Frameworks

• 2.2.2.1 Real-World to Number Sentence
• 2.2.2.2 Number Sentence to Real World

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What is a Number Sentence?

• Teaching Strategies, Tactics, and Methods

A number sentence is an array of numbers and symbols. Also referred to as a “sum” or “problem,” number sentences are a familiar way of arranging questions in K-5 math.

Kids must learn this early, as it is how most of the work in their math lessons will look.

“Number sentence” is the term used in K-5-level math teaching in countries such as the U.S., Canada, the UK, New Zealand, South Africa, and Australia.

Here are some examples of number sentences:

Addition number sentence: 7 + 5 = 12

Subtraction number sentence: 44 – 10 = 34

Multiplication number sentence: 5 x 4 = 20

Division number sentence: 35 ÷ 7 = 5

When are number sentences introduced?

Typically, children will start learning to write and solve number sentences in first grade. They will likely begin by using objects such as counters and small toys to help them understand the value of numbers.

Children also need to be able to turn word problems into number sentences to understand the question. For example:

Steve has $10 and spends $4.50 on his lunch. How much does he have left?

The number sentence is 10 – 4.50 = 5.50

So a child can work out that Steve has $5.50 left using a number sentence.

They are essential for a child to learn early on to develop their math skills, as this is what most of the math problems they’ll be solving will look like.

Number sentences for kids

Here are ten differentiated number sentences for your children to try out. They start friendly and accessible but get harder and harder as you go on. How many can your class solve?

• 3 – 2 = ?
• 6 – ? = 2
• 10 – ? = 3
• 25 + 12 = ?

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Strategy: Write a Number Sentence

Math Problem Solving Strategy: Write a Number Sentence to Solve a Problem

This is another free resource for teachers from The Curriculum Corner.

Looking to help your students learn to write a number sentence to solve a problem?

This math problem solving strategy can be practiced with this set of resources.

Math Problem Solving Strategies

This is one in a series of resources to help you focus on specific problem solving strategies in the classroom.

Within this download, we are offering you a range of word problems for practice.

Each page provided contains a single problem solving word problem.

Below each story problem you will find a set of four steps for students to follow when finding the answer.

This set will focus on the write a number sentence strategy for math problem solving.

What are the 4 problem solving steps?

After carefully reading the problem, students will:

• Step 1:  Circle the math words.
• Step 2:  Ask yourself: Do I understand the problem?
• Step 3:  Solve the problem using words and pictures below.
• Step 4:  Share the answer along with explaining why the answer makes sense.

Write a Number Sentence to Solve a Problem Word Work Questions

The problems within this post are meant to help students solve problems by writing a number sentence.  These problems are designed to be used with first, second or third grade math students.

Within this collection you will find two variations of each problem.

You will easily be able to create additional problems using the wording below as a base.

The problems include the following selections:

• Cookies – easy addition
• Coin Collection – addition with regrouping
• Jewelry – addition with regrouping
• Making Cards – easy subtraction
• Beads for Bracelets – subtraction without regrouping
• Toy Cards – subtraction with regrouping
• Hot Chocolate – easy multiplication
• Pencils – one-digit times two-digit multiplication
• Legos – two-digit times two-digit multiplication

You can download this set of Write a Number Sentence to Solve a Problem here:

Problem Solving

You might also be interested in the following problem solving resources:

• Drawing Pictures to Solve Problems
• Addition & Subtraction Word Problem Strategies
• Fall Problem Solving
• Winter Problem Solving
• Spring Problem Solving
• Summer Problem Solving

As with all of our resources, The Curriculum Corner creates these for free classroom use. Our products may not be sold. You may print and copy for your personal classroom use. These are also great for home school families!

You may not modify and resell in any form. Please let us know if you have any questions.

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Monday 27th of January 2020

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Great activities for understanding how to solve word problems.

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Number Sentences

Grab your binoculars to take an expedition in this Number Sentences song, where Edwin learns that Number Sentences are equations - or mathematical sentences - that show statements using numbers and math symbols.

His Number Sentences journey takes him from the jungles of Borneo to the ice shelves off the coast of Greenland on his ship, the SS Mathematica. The only thing Edwin loves as much as the wildlife he encounters along the way is his newfound ability to communicate what he has seen to his friends using Number Sentences.

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• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• One-Step Addition
• One-Step Subtraction
• One-Step Multiplication
• One-Step Division
• One-Step Decimals
• Two-Step Integers
• Two-Step Add/Subtract
• Two-Step Multiply/Divide
• Two-Step Fractions
• Two-Step Decimals
• Multi-Step Integers
• Multi-Step with Parentheses
• Multi-Step Rational
• Multi-Step Fractions
• Multi-Step Decimals
• Solve by Factoring
• Completing the Square
• Quadratic Formula
• Biquadratic
• Logarithmic
• Exponential
• Rational Roots
• Floor/Ceiling
• Equation Given Roots
• Newton Raphson
• Substitution
• Elimination
• Cramer's Rule
• Gaussian Elimination
• System of Inequalities
• Perfect Squares
• Difference of Squares
• Difference of Cubes
• Sum of Cubes
• Polynomials
• Distributive Property
• FOIL method
• Perfect Cubes
• Binomial Expansion
• Negative Rule
• Product Rule
• Quotient Rule
• Expand Power Rule
• Fraction Exponent
• Exponent Rules
• Exponential Form
• Logarithmic Form
• Absolute Value
• Rational Number
• Powers of i
• Complex Form
• Partial Fractions
• Is Polynomial
• Leading Coefficient
• Leading Term
• Standard Form
• Complete the Square
• Synthetic Division
• Linear Factors
• Rationalize Denominator
• Rationalize Numerator
• Identify Type
• Convergence
• Interval Notation
• Pi (Product) Notation
• Boolean Algebra
• Truth Table
• Mutual Exclusive
• Cardinality
• Caretesian Product
• Age Problems
• Distance Problems
• Cost Problems
• Investment Problems
• Number Problems
• Percent Problems
• Addition/Subtraction
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• Dice Problems
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• Pre Calculus
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• Conversions

Most Used Actions

Number line.

• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

word-problems-calculator

• Middle School Math Solutions – Inequalities Calculator Next up in our Getting Started maths solutions series is help with another middle school algebra topic - solving...

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Intervention based on science of reading, math boosts comprehension, word problem-solving skills

English learners with math difficulty showed improvement following culturally-responsive training.

New research from the University of Kansas has found an intervention based on the science of reading and math effectively helped English learners boost their comprehension, visualize and synthesize information, and make connections that significantly improved their math performance.

The intervention, performed for 30 minutes twice a week for 10 weeks with 66 third-grade English language learners who displayed math learning difficulties, improved students' performance when compared to students who received general instruction. That indicates emphasizing cognitive concepts involved in the science of reading and math are key to helping students improve, according to researchers.

"Word problem-solving is influenced by both the science of reading and the science of math. Key components include number sense, decoding, language comprehension and working memory. Utilizing direct and explicit teaching methods enhances understanding and enables students to effectively connect these skills to solve math problems. This integrated approach ensures that students are equipped with necessary tools to navigate both the linguistic and numerical demands of word problems," said Michael Orosco, professor of educational psychology at KU and lead author of the study.

The intervention incorporates comprehension strategy instruction in both reading and math, focusing and decoding, phonological awareness, vocabulary development, inferential thinking, contextualized learning and numeracy.

"It is proving to be one of the most effective evidence-based practices available for this growing population," Orosco said.

The study, co-written with Deborah Reed of the University of Tennessee, was published in the journal Learning Disabilities Research and Practice .

For the research, trained tutors developed the intervention, developed by Orosco and colleagues based on cognitive and culturally responsive research conducted over a span of 20 years. One example of an intervention session tested in the study included a script in which a tutor examined a word problem that explained a person made a quesadilla for his friend Mario, giving him one-fourth of it, then needed to students to determine how much remained.

The tutor first asked students if they remembered a class session in which they made quesadillas, what shape they were and demonstrated concepts by drawing a circle on the board, dividing it into four equal pieces, having students repeat terms like numerator and denominator, and explaining that when a question asks how much is left, subtraction is required. The students also collaborated with peers to practice using important vocabulary in sentences. The approach both helps students learn and understand mathematical concepts while being culturally responsive.

"Word problems are complex because they require translating words into mathematical equations, and this involves integrating the science of reading and math through language concepts and differentiated instruction," Orosco said. "We have not extensively tested these approaches with this group of children. However, we are establishing an evidence-based framework that aids them in developing background knowledge and connecting it to their cultural contexts."

Orosco, director of KU's Center for Culturally Responsive Educational Neuroscience, emphasized the critical role of language in word problems, highlighting the importance of using culturally familiar terms. For instance, substituting "pastry" for "quesadilla" could significantly affect comprehension for students from diverse backgrounds. Failure to grasp the initial scenario can impede subsequent problem-solving efforts.

The study proved effective in improving students' problem-solving abilities, despite covariates including an individual's basic calculation skills, fluid intelligence and reading comprehension scores. That finding is key as, while ideally all students would begin on equal footing and there were little variations in a classroom, in reality, covariates exist and are commonplace.

The study had trained tutors deliver the intervention, and its effectiveness should be further tested with working teachers, the authors wrote. Orosco said professional development to help teachers gain the skills is necessary, and it is vital for teacher preparation programs to train future teachers with such skills as well. And helping students at the elementary level is necessary to help ensure success in future higher-level math classes such as algebra.

The research builds on Orosco and colleagues' work in understanding and improving math instruction for English learners. Future work will continue to examine the role of cognitive functions such as working memory and brain science, as well as potential integration of artificial intelligence in teaching math.

"Comprehension strategy instruction helps students make connections, ask questions, visualize, synthesize and monitor their thinking about word problems," Orosco and Reed wrote. "Finally, applying comprehension strategy instruction supports ELs in integrating their reading, language and math cognition… Focusing on relevant language in word problems and providing collaborative support significantly improved students' solution accuracy."

• Learning Disorders
• K-12 Education
• Educational Psychology
• Intelligence
• Special education
• Problem solving
• Developmental psychology
• Child prodigy
• Intellectual giftedness
• Lateral thinking

Story Source:

Materials provided by University of Kansas . Original written by Mike Krings. Note: Content may be edited for style and length.

Journal Reference :

• Michael J. Orosco, Deborah K. Reed. Supplemental intervention for third-grade English learners with significant problem-solving challenges . Learning Disabilities Research & Practice , 2024; 39 (2): 60 DOI: 10.1177/09388982241229407

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Despite a fortified border, migrants will keep coming, analysts agree. Here's why.

Sergio Martínez-Beltrán

Border Patrol picks up a group of people seeking asylum from an aid camp near Sasabe, Arizona, on Wednesday, March 13, 2024. Justin Hamel/Bloomberg via Getty Images hide caption

Border Patrol picks up a group of people seeking asylum from an aid camp near Sasabe, Arizona, on Wednesday, March 13, 2024.

The U.S. southern border is as fortified as ever and Texas is carrying out its own enforcement to stop people from crossing illegally, yet observers and analysts agree on this: migrants not only will continue to come, but their numbers will likely increase in the coming months.

The expected surge can be attributed not only to seasonal migration patterns, but an increase of people displaced by war, poverty, and climate factors in all continents.

And why do these analysts say this?

They keep a close eye on the Darién Gap in Panama and the borders between Central American countries, two key points to gauge the number of people venturing up north.

"In most countries (outward) migration has increased ... particularly in Venezuela, and that's not really reflected yet in the U.S. numbers," said Adam Isacson, an analyst of border and migration patterns at the Washington Office on Latin America, a nonpartisan research and advocacy organization based in Washington D.C.

Despite Mexico's cracking down on migrants, Isacson said people are still making their way up north, even if they need to pause for months at different points during their journey.

"There must be a huge number of people from Venezuela bottled up in Mexico right now," he said.

The Darién Gap serves as a good barometer for migration flows.

This 100-mile-long tropical jungle between Colombia and Panama has claimed the lives of hundreds of migrants, according to a report from the Migration Policy Institute, a Washington, D.C.-based think tank.

Yet the dangers at this jungle are not a deterrent, said Ariel Ruiz Soto, a senior policy analyst with this organization. The majority of people migrating are from Venezuela.

"The reason why I referred to Venezuelans in particular is because they represent a key challenge for removals from Mexico and from the United States to Venezuela," Ruiz Soto said.

Mexico and the U.S. had been flying Venezuelan migrants back to the South American country . However, earlier this year, Venezuelan President Nicolás Maduro stopped accepting flights from the U.S. in response to economic sanctions imposed by the Biden administration.

Panama reported a 2% increase in crossings through the Darién Gap in February compared to the previous month.

Aerial view showing migrants walking through the jungle near Bajo Chiquito village, the first border control of the Darien Province in Panama, on September 22, 2023. LUIS ACOSTA/AFP via Getty Images hide caption

Aerial view showing migrants walking through the jungle near Bajo Chiquito village, the first border control of the Darien Province in Panama, on September 22, 2023.

What the numbers show

Analysts are projecting the increase in the remaining months of the fiscal year, even though U.S. Customs and Border Protection reported a 2.2% decrease in encounters with migrants along the Southern border in March. An encounter is every time a migrant is picked up by immigration authorities.

These numbers are consistent with cyclical patterns of illegal crossings that dip in the winter months, followed by more migrants attempting to get to the U.S. as warm weather arrives, said Ruiz Soto.

In a statement, CBP Spokesperson Erin Waters said the agency remains vigilant to "continually shifting migration patterns" amid "historic global migration."

Waters said the agency has also been partnering with Mexico to curb the flow of people migrating to the U.S.

Mexico has commissioned its National Guard to patrol its borders with Guatemala and the U.S.

"CBP continues to work with our partners throughout the hemisphere, including the Government of Mexico, and around the world to disrupt the criminal networks who take advantage of and profit from vulnerable migrants," Waters said.

Where are migrants crossing the border?

For the last few months, more migrants are attempting to cross through Arizona instead of Texas, according to CBP.

In 2023, the El Paso and Del Rio sector in Texas saw more crossings than any other place across the 2,000-mile Southern border. But this year the Tucson sector in Arizona has seen a 167% increase in crossings, more than any other.

Tiffany Burrow, operations director at Val Verde Border Humanitarian Coalition, an assistance organization for newly border crossers in Del Rio, said she has seen the shift.

"It's empty," Burrow said, pointing to her organizations' office. "There are no migrants."

In March, she helped only three migrants after they were released by CBP pending their court date. In December, they helped 13,511 migrants.

Burrow said that's how migration works — it ebbs and flows.

"We have to be ready to adapt," Burrow said.

Texas Department of Safety Troopers patrol on the Rio Grande along the U.S.-Mexico border. Eric Gay/AP hide caption

Texas Department of Safety Troopers patrol on the Rio Grande along the U.S.-Mexico border.

Texas' role

Burrow and other immigrant advocates are closely observing Texas' ramping up of border enforcement.

In 2021 Gov. Greg Abbott launched Operation Lone Star initiative and deployed the Texas National Guard. Last year the state started lining up razor wire in sections of the Rio Grande.

Texas is also asking the courts to be allowed to implement a law passed last year by the Republican-controlled legislature, known as SB4, which requires local and state police to arrest migrants they suspect are in the country illegally.

It might be too early to know if all these efforts will have an impact on migration patterns, analysts said, considering that Texas saw the highest number of illegal crossings last year.

But, Mike Banks, special advisor on border matters to Abbott, said the state's efforts are fruitful.

Texas has spent over $11 billion in this initiative.

"The vast majority of the United States' southern border is in Texas, and because of Texas' efforts to secure the border, more migrants are moving west to illegally cross the border into other states," said Mike Banks in a statement to NPR.

Ruiz Soto, from the Migrant Policy Institute, said the impact of Texas' policies on arrivals "is likely to be minimal over the long term."

Carla Angulo-Pasel, an assistant professor who specializes in border studies and international migration at the University of Texas at Rio Grande Valley, said that even with Texas' policies in place, migrants are likely to continue to cross.

"You can't claim, as much as I think Gov. Abbott wants to claim, that Operation Lone Star is going to somehow mean that you're going to see less numbers in Texas because that hasn't held true," Angulo-Pasel said. "We could also argue that things are going to progressively get more and more as the spring months progress."